He was born in London, England on the 3rd March 1852 to Edward Palmer and his wife, Caroline. He was educated at Lancing College, Sussex, before joining the financial department of the Indian government in 1871. He became assistant comptroller-general before leaving India for Egypt in 1855. Here, he suceeded Sir Gerald Fitz-gerald as director-general of accounts in Egypt. He was made CMG in 1888 and a year later succeeded Sir Edgar Vincent as financial advisor to the Khedive.
Sir Elwin was appointed first governor to the newly created National Bank of Egypt in 1898. In 1902 he was appointed president of the Agricultural Bank of Egypt, a subsidiary of the National Bank. He was made KCMG in 1892 and KCB in 1897 whilst also being awarded the grand cordens of the Order of Osmanie and the Order of Medjidie.
He died in Cairo on 28th January 1906.